Beamforming in Wireless Relay Networks

Transcription

1 1 Beaforng n Wreess Reay Networks YINDI JING AND HAMID JAFARKHANI Departent of Coputer Scence and Eectrca Engneerng Unversty of Caforna Irvne Irvne CA 9697 Abstract Ths paper s on reay beaforng n wreess networks n whch the recever has perfect nforaton of a channes and each reay knows ts own channes Instead of the coony used tota power constrant on reays and the transtter we use a ore practca assupton that every node n the network has ts own power constrant A twostep apfy-and-forward protoco wth beaforng s used n whch the transtter and reays are aowed to adaptvey adust ther transt power and drectons accordng to avaabe channe nforaton The opta beaforng probe s soved anaytcay The copexty of fndng the exact souton s near n the nuber of reays Our resuts show that the transtter shoud aways use ts axa power and the opta power used at a reay s not a bnary functon It can take any vaue between zero and ts axu transt power Aso nterestngy ths vaue depends on the quaty of a other channes n addton to the reay s own ones Despte ths coupng fact dstrbutve strateges are proposed n whch wth the ad of a ow-rate broadcast fro the recever a reay needs ony ts own channe nforaton to peent the opta power contro Suated perforance shows that network beaforng acheves fu dversty and outperfors other exstng schees Index Ters Reay network beaforng cooperatve dversty I INTRODUCTION It s we-known that due to the fadng effect the transsson over wreess channes suffers fro severe attenuaton n sgna strength Perforance of wreess councaton s uch worse than that of wred councaton For the spest pont-to-pont councaton syste whch s coposed of one transtter and one recever ony the use of utpe antennas can prove the syste capacty and reabty Space-te codng and beaforng are aong the ost successfu technques deveoped for utpe-antenna systes durng the ast few decades [1] [] However n any stuatons due to the ted sze and processng power t s not practca for soe users especay sa wreess obe devces to have utpe antennas Thus recenty wreess network councaton s attractng ore and ore attenton A arge aount of effort has been gven to prove the councaton by havng dfferent users n a network cooperate Ths proveent s conventonay addressed as cooperatve dversty and the technques are addressed as cooperatve schees Many cooperatve schees have been proposed n the terature [3] [1] Soe assue channe nforaton at the recever but not the transtter and reays for exape the noncoherent apfy-and-forward protoco n [8] [9] and dstrbuted spacete codng n [10] Soe assue channe nforaton at the recevng sde of each transsson for exape the decodeand-forward protoco n [8] [1] and the coded-cooperaton n [13] Soe assue no channe nforaton at any node for exape the dfferenta transsson schees proposed ndependenty n [14] [16] The coherent apfy-and-forward schee n [9] [11] assues fu channe nforaton at both reays and the recever But ony channe phase nforaton s used at reays In a these cooperatve schees the reays aways cooperate on ther hghest powers None of the above poneer work aow reays to adust ther transt powers adaptvey accordng to channe agntude nforaton and ths s exacty the concern of ths paper There have been severa papers on reay networks wth adaptve power contro In [] [3] the outage capacty of networks wth a snge reay and perfect channe nforaton at a nodes was anayzed Both work assue a tota power constrant on the reay and the transtter A decode-andforward protoco s used at the reay whch resuts n a bnary power aocaton between the reay and the transtter In [4] perforance of networks wth utpe apfy-and-forward reays and an aggregate power constrant was anayzed A dstrbutve schee for the opta power aocaton s proposed n whch each reay ony needs to know ts own channes and a rea nuber that can be broadcasted by the recever Another reated work on networks wth one and two apfyand-forward reays can be found n [5] In [6] outage nzaton of snge-reay networks wth ted channenforaton feedback s perfored It s assued that there s a ong-ter power constrant on the tota power of the transtter and the reay In [38] we anaytcay sove the reay power contro probe for networks wth two reays and provde an teratve agorth for networks wth ore than two reays In ths paper we consder networks wth a genera nuber of apfy-and-forward reays and we assue a separate power constrant on each reay and the transtter The anaytca power contro souton for networks wth any nuber of reays s found Due to the dfference n the power assuptons copared to [4] anayss of ths new ode s ore dffcut and totay dfferent resuts are obtaned For utpe-antenna systes when there s no channe nforaton at the transtter space-te codng can acheve fu dversty [1] If the transtter has perfect or parta channe nforaton perforance can be further proved through beaforng snce t takes advantage of the channe nforaton both drecton and strength at the transtter sde to obtan hgher receve SNR [] Wth perfect channe nforaton or hgh quaty channe nforaton at the transtter one-densona beaforng s proved opta [] [7] [8] The ore practca utpe-antenna systes wth parta channe nforaton at the transtter channe statstcs or quantzed nstantaneous channe nforaton are

2 s Fg 1 transtter f f 1 f Wreess reay network R r reays r r R t t t 1 R g g 1 g R recever aso anayzed extensvey [9] [33] In any stuatons approprate cobnaton of beaforng and space-te codng outperfors ether one of the two schees aone [34] [37] In ths paper we w see sar perforance proveent n networks usng network beaforng over dstrbuted spacete codng and other exstng schees such as best-reay seecton and coherent apfy-and-forward We consder networks wth one par of transtter and recever but utpe reays The recever knows a channes and every reay knows ts own channes perfecty A twostep apfy-and-forward protoco s used where n the frst step the transtter sends nforaton and n the second step the reays transt Frst t s proved that the transtter shoud aways use ts fu power Then the opta reay beaforng vector s found anaytcay The exact souton can be obtaned wth a copexty that s near n the nuber of reays To perfor network beaforng we propose two dstrbutve strateges n whch a reay needs ony ts own channe nforaton and a ow-rate feedback fro the recever whch s coon to a reays Suaton shows that network beaforng outperfors other exstng schees We shoud carfy that ony apfy-and-forward s consdered here For decode-and-forward the resut ay be dfferent and t depends on the detas of the codng schees The paper s organzed as foows In the next secton the reay network ode and the dea of network beaforng are ntroduced In Secton III the network beaforng probe s soved anaytcay Then n Secton IV we propose two schees to peent the opta reay beaforng dstrbutvey n networks Suated perforance of network beaforng and ts coparson to other schees are shown n Secton V Secton VI contans the concuson and severa future drectons Notaton: For a atrx A A T and A denote the transpose and Hertan of A ndcates the nner product ndcates the -nor P ndcates the probabty a denotes the th coordnate of vector a and a 1 k denotes the k- densona vector [ a 1 a k ] T If ab are two R- densona vectors a b eans a b for a 1R 0 R s the R-densona vector wth a zero entres II NETWORK MODEL AND PROBLEM STATEMENT Consder a reay network wth one transt-and-receve par and R reays as depcted n Fg 1 Every reay has one snge antenna whch can be used for both transsson and recepton Denote the channe fro the transtter to the th x reay as f and the channe fro the th reay to the recever as g We assue that the th reay knows ts own channes f and g and the recever knows a channes f 1 f R and g 1 g R Our resuts are vad for any channe statstcs We assue that for each transsson the powers used at the transtter and the th reay are no arger than P 0 and P respectvey Note that n ths paper ony short-ter power constrant s consdered that s there s an upper bound on the average transt power of each node for each transsson A node cannot save ts power to favor transssons wth better or worse channe reazatons We use a two-step apfy-and-forward protoco Durng the frst step the transtter sends α 0 P0 s The nforaton sybo s s seected randoy fro the codebook S If we noraze t as E s 1 the average power used at the transtter s α 0P 0 The th reay receves r α 0 P0 f s v where v s the nose at the th reay We assue that t s CN01 Durng the second step the th reay sends P t α e θ r 0 f P 0 Note that n contrast wth the tradtona apfy-and-forward protoco n whch P t r 0 f P 0 the coeffcent α and the ange θ are used to adust the transt power and drecton of Reay The average transt power of the th reay can be cacuated to be α P The receved sgna at the recever can thus be cacuated to be xα 0 P0 R α e θ f g P 0 f P 0 s α e θ g P 0 f P 0 v w 1 where w s the nose at the recever It s aso assued to be CN0 1 The tota average power n transttng one sybo s thus no arger than R 0 P Equaton 1 can be rewrtten as where x α 0 P0 hxs v x [ α 1 e θ1 α R e θr ] T s the reay beaforng vector [ f h 1g 1 P1 1α 0 f 1 P 0 ] f Rg R PR 1α 0 f R P 0 s the equvaent channe vector and α g e θ P v v w 0 f P 0 s the equvaent nose Our network beaforng desgn s thus the desgn of α 0 and x or equvaenty θ 1 θ R and α 0 α 1 α R such

3 3 that the error rate of the network s the saest Ths s equvaent to axze the receve SNR The syste equaton has the sae structure as the syste equaton of beaforng n a utpe-antenna syste wth R transt antennas and one receve antenna The ater has been soved anaytcay and the opta beaforng vector s x opt h / h However there are two an dfferences between reay networks and utpe-antenna systes Frst n networks the reays have ony nosy versons of the transt sgna Thus the nforaton s corrupted by noses fro both the recever and the reays Second transssons n reay networks usuay take ore than one hop whch resuts n totay dfferent SNR or capacty cacuatons We can see that both the equvaent channe vector h and the equvaent nose v n have ore copcated natures than those n utpe-antenna systes Aso the nose v depends on the beaforng vector x Thus the network beaforng probe s uch ore dffcut III ANALYTICAL BEAMFORMING RESULT In ths secton we present the anaytca network beaforng resut Snce v s are d CN01 the vaues of the anges θ have no effect on the nose power To axze the sgna power the R coponents of the frst ter n the rght hand sde of 1 shoud add coherenty Thus an opta choce of the anges s θ arg f arg g That s atch fters shoud be used at reays to cance the phases of ther channes and for a bea at the recever Wth ths choce we have xα 0 P0 R α f g P 0 f P 0 s α g P 0 f P 0 e arg f v w 3 What s eft of our probe s the opta power contro e the optzaton of α 0 α 1 α R Ths s aso the an contrbuton of our work Fro 3 the receve SNR can be cacuated to be α 0 P 0 R 1 R α f g P 1α 0 f P 0 α g P 1α 0 f P 0 It s an ncreasng functon of α 0 Therefore the transtter shoud aways use ts axa power e α 0 1 The receve SNR s thus: P 0 R α f g P 1 f P 0 1 R α g P 1 f P 0 To show the portance of adaptve reay power contro we gve a spe two-reay network exape found by coputer suaton Let P 0 P 1 P 10dB and the channe vaues are f f 1640 g and g 0797 In Tabe I the receve SNR vaues of dfferent schees are shown as we as ther vaues of α 1 α We can see that TABLE I A RELAY POWER CONTROL EXAMPLE FOR A -RELAY NETWORK α 1 α receve SNR opta power contro no power contro better reay seecton n ths exape the receve SNR of network beaforng s about 68% and 5% arger than those of no reay power contro both reays use fu power and the better reay seecton respectvey Now et us work on the SNR axzaton We frst ntroduce soe notaton to hep the presentaton Wth a sght abuse of notaton we defne our vector of unknown power coeffcents as Aso defne b a [ [ x [ α 1 α R ] T f 1g 1 P 1 1 f1 P 0 g 1 P 1 1 f1 P 0 f ] Rg R P R 1 fr P 0 g ] R P R 1 fr P 0 and A dag{a} where dag{a} s the dagona atrx whose th dagona entry s a Let Λ { v R R 0 R v a } and for 1R and 1 1 < < k R et Λ 1 k { v R k 0 k v a 1 k } Wth the transforaton y Ax or equvaenty x A 1 y we have where bx SNR P 0 1 Ax P cy 0 1 y c A T b [ f 1 f R ] T The receve SNR optzaton probe s thus equvaent to ax y cy 1 y st y Λ 4 The dffcuty of the probe es n the shape of the feasbe set If y s constraned on a hypersphere that s y r the souton s obvous at east geoetrcay Gven that y r cy 1 y r c 1 r cos ϕ where ϕ s the ange between c and y The opta souton shoud be the vector whch has the saest ange wth c Ths provdes us an dea to sove the probe For any gven r we frst fx the ength of y as r and fnd the opta drecton of y Then we can sove the probe by fndng the opta ength The feasbe nterva for the ength of y s [0 a ] Due to the shape of the regon we need to decopose ths nterva nto R saer ones Ths requres a reay orderng as foows

4 4 Snce Pa > 0 1 and Pc > 0 1 we assue that a > 0 and c > 0 Defne φ φf g P c f 1 f P 0 a g 5 P for 1 R and for the sake of presentaton defne φ R1 0 Order φ as φ τ1 φ τ φ τr φ τr1 6 τ 1 τ τ R τ R1 s thus an orderng of 1RR 1 and τ R1 R 1 Defne r 00 r 1φ 1 τ 1 c r r R 1 Õ Ú φ Ø φ Ú Õ φ τ 1 c τ τ R a τ 1 τ c τ τ R a τ 1 R τ R 1 c τr 1 τ R R 1 r R Ø φ τ R c τr a τ a Ú Ø φ τ c τ3 τ R a τ a τ Ú R 1 Ø φ τ R 1 c τr Snce φ τ 1 φ τ we have r 1 r for 1R Thus the nterva [0 a ] s decoposed nto the foowng R ntervas: [0 a ] [r 0 r 1 ] [r 1 r ] [r R 1 r R ] Let Γ [r r 1 ] The optzaton probe n 4 s equvaent to ax ax 1 R r Γ 1 r ax cy y r Γ y Λ It can be soved through the foowng three steps Step 1: For a gven ength r n Γ fnd the opta drecton of y Λ e sove the nner product axzaton probe: ax cy 7 y r Γ y Λ Step : Fnd the opta ength nsde Γ e sove the th subprobe: y 1 ax r Γ 1 r ax cy 8 y r Γ y Λ by sovng c ax r Γ 1 r Step 3: Search over the R soutons of the R subprobes and fnd the opta souton for the whoe probe The foowng eas and theore soves each step and thus the whoe probe Lea 1: a for τ 1 τ or equvaenty z τ r 1τ a τ1τ Proof: We prove ths ea by contradcton Assue < a for soe {τ 1 τ } We frst show that that a τ there exsts an { τ R } such that Assue that zr c have < zr c c c < c a c Ú Ø Ú Ø Ú τ R a τ R 1 c < zr c for a { τ R } We c φ 1 Thus 1 Ø φ c τ1 τ R Ú Ø φ τ c τ1 τ R r c τ φ z τ r a τ a τ because of 6 Ths contradcts r Γ Thus there exsts an { τ R } such that zr c < zr c Defne another vector z as z δz r where 0 < δ < n c δ and z 1 c c 1 c z z r δ z r c for a Snce we have assued that < a and have ust proved that c < zr c such δ s achevabe To contradct the assupton that s the opta t s enough to prove the foowng two tes: 1 z s a feasbe pont e z r and z Λ c < cz Fro the defnton of z we have z z z δ r z r δz δ Snce 0 < δ < a we have 0 < z < a Aso snce 0 < δ < r z we can easy prove that z r δz δ > 0 and z < zr a Thus z Λ The frst te has been proved For the 1 second te snce δ < c 1 c c zr c we c }

5 5 have 1 c c c c δ < c c c c c δ < c z δ c c δ < zr δ c c c z c z < 0 c < cz z r c c δ δ z Lea : λ r c for τ R or equvaenty z τ r 1τ R λ r c τ1τ R where r λ r a τ c τ1τ R Proof: Fro Lea 1 a for τ 1 τ Thus 7 can be wrtten as ax y r Γ y Λ b τ It s obvous that a τ c τ c τ1 τ R y τ1 τ R ax È c τ1τr r a τ r Γ y τ R Λ τ R y τ R yτ1τr c τ1τ R y τ1τ R c τ1τ R λ r c τ1τ R for a y τ1τ R r a τ In other words to axze the nner product y τ1τ R shoud have the sae drecton as c τ1τ R Thus we ony need to show that ths drecton s feasbe for r Γ Ths s equvaent to show that λ r c τ1τ R Λ τ1τ R for any r Γ We can easy prove that r Γ λ r Ω [ ] where Ω φ 1 τ φ 1 for 0R 1 Thus for any r Γ and τ R we have 0 λ r c φ 1 c φ 1 c a Hence λ r c τ1τ R Λ τ1τ R Cobnng Lea 1 and Lea we have and thus { a τ 1 τ λ r c τ R 9 ax cy b τ λ r c τ1τ R 10 y r Γ y Λ We have soved the nner product axzaton probe of Step 1 The soutons of the R subprobes can thus be obtaned Lea 3: For 1R 1 defne λ 1 a τ b τ The souton of Subprobe 0 s y 0 φ 1 τ 1 c For 1R 1 the souton of Subprobe y s defned δ as { a y { } τ 1 τ n λ φ 1 11 c τ R Proof: Fro 10 Subprobe s equvaent to the foowng 1-densona optzaton probe: b τ c τ1τ R λ ax λ Ω 1 a τ c τ1τ R λ 1 c When 0 1 s equvaent to ax 4 λ λ Ω 1 c λ Snce c 4 λ 1 c λ s an ncreasng functon of λ ts axu s at λ φ 1 τ 1 For 1R 1 defne We have ξ λ ξ λ ξ b τ c τ1τ R λ 1 a τ c τ1τ R λ b τ c τ1τ R λ c τ1τ R 1 a τ c τ1τ R λ 1 a τ b τ λ Thus λ > 0 f λ < λ and ξ λ < 0 f λ > λ So f λ φ 1 the opta souton s reached at λ λ Otherwse the opta souton s reached at λ φ 1 Fro 9 Subprobe s soved at y as defned n 11 Now we are ready for souton of the reay power contro probe n 4 Theore 1: Defne x as { x 1 τ1 τ 13 λ φ τ R The souton of the SNR optzaton s x 0 where 0 s the saest such that λ < φ 1 Proof: Frst snce φ R1 0 we have λ R < φ 1 τ R1 φ 1 R1 Thus 0 exsts Aso snce λ 0 < φ 1 τ 0 1 and φ τ decreases wth we have x 0 1 for τ 01τ R Ths eans that x 0 s n the feasbe regon of the optzaton probe Denote ηy cy 1 y Note that y 0 r 1 Snce r 1 Γ 1 y 0 s aso a feasbe pont of Subprobe 1 Thus η y 0 η y 1 due to the optaty of y 1 n Subprobe 1 Ths eans that there s no need to consder Subprobe 0 For 1R f λ φ 1 y { a τ 1 τ φ 1 c τ R

6 6 and y φ cτ1τ R a τ r 1 1 Snce r 1 Γ 1 y s a feasbe pont of Subprobe 1 Thus η y η y 1 due to the optaty of y 1 n Subprobe 1 Ths eans that there s no need to consder Subprobe 1 Thus we ony need to check those y s wth λ < φ 1 and fnd the one that resuts n the argest receve SNR Fro the defnton n 13 ths s the sae as to check those x s wth λ < φ 1 Now we prove that λ 1 < φ 1 τ f λ < φ 1 Frst fro λ < φ 1 we have 1 a τ b τ < φ 1 Snce a b τ1 φ 1 we can prove easy that λ a τ 1 b τ 1 a τ a b τ b τ1 <φ 1 < φ 1 τ Thus we ony need to check those x s for 0 R and fnd the one causng the argest receve SNR Fro prevous dscusson 0 1 Defne SNR bx Now we prove that SNR 1 Ax > SNR 1 for 0 R Fro the proof of Lea 3 we have b τ c τ1τ R λ SNR 1 a τ c τ1τ R λ c τ 1 a τ 1 1 b τ c τ 1 a τ λ b τ SNR 1 b a 1 1 b τ SNR 1 >SNR 1 a τ 1 1 a τ 1 a τ a 1 φ1 λ 1 1 a 1 τ Thus the opta power contro vector that axzes the receve SNR s x 0 IV DISTRIBUTED SCHEMES FOR NETWORK BEAMFORMING It s natura to expect the power contro at reays to undergo an on-or-off scenaro: a reay uses ts fu power f ts channes are good enough and otherwse not to cooperate at a Our resut shows otherwse The opta power used at a reay can be any vaue between 0 and ts axa power In any stuatons a reay shoud use a fracton of ts power whose vaue s deterned not ony by ts own channes but a others as we Ths s because every reay has two effects on the transsson For one t heps the transsson by forwardng the nforaton whe for the other t hars the transsson by forwardng nose as we Its transt power has a nonnear effect on the powers of both the sgna and the nose whch akes the optzaton souton not an on-or-off one not a decouped one and n genera not even a dfferentabe functon of channe coeffcents As shown n Theore 1 the fracton of power used at reay satsfes α 1 for τ 1 τ 0 and α λ 0 φ for τ 01τ R Thus the 0 reays whose φ s are the argest use ther fu power Snce 0 1 there s at east one reay that uses ts fu power Ths tes us that the reay wth the argest φ aways uses ts fu power The reanng R 0 reays whose φ s are saer ony use parts of ther power For τ 01τ R the power used at the th reay s α P λ 0 φ P λ 0 f /g 1 f P 0 whch s proportona to f /g 1 f P 0 snce λ0 s a constant for each channe reazaton Athough P does not appear expcty n the forua t affects the decson of whether the th reay shoud use ts fu power Actuay n deternng whether a reay shoud use ts axa power not ony do the channe coeffcents and power constrant at ths reay account but aso a other channe coeffcents and power constrants The power constrant of the transtter P 0 pays a ro as we Due to these speca propertes of the opta power contro souton t can be peented dstrbutvey wth each reay knowng ony ts own channe nforaton In the foowng we propose two dstrbuted strateges One s for networks wth a sa nuber of reays and the other s ore econoca n networks wth a arge nuber of reays The recever whch knows a channes can sove the power contro probe When the nuber of reays R s sa the recever broadcasts R bts 1 R and the coeffcent λ 0 The bnary sybo s to ndcate whether Reay shoud use ts fu power If 1 Reay uses ts fu power to transt durng the second step Otherwse t uses power λ 0 f /g 1 f P 0 The nuber of requred feedback bts s RB 1 where B 1 s the aount of bts needed n broadcastng the rea nuber λ 0 Ths nuber ncreases neary n the network sze The other schee s to have the recever broadcast two rea nubers: λ 0 and a rea nuber d that satsfes φ τ0 > d > φ τ0 1 Reay cacuates ts own φ If φ > d Reay uses ts fu power Otherwse t uses power λ 0 f /g 1 f P 0 The nuber of bts needed for ths

7 7 feedback s B 1 Thus when R s arge ths strategy needs ess bts of feedback copared to the frst one Networks wth an aggregate power constrant P on reays were anayzed n [4] In that case wth the sae notaton n Secton III P P and R 1 α 1 The opta reay power aocaton souton s α f g 1 f P 0 f P 0 g P1 R f g 1 f P 0 f P 0 g P1 α s a functon of ts own channes f g ony and an extra R f coeffcent c g 1 f P 0 f P 0 g P1 whch s the sae for a reays Therefore ths power aocaton can be done dstrbutvey wth the extra knowedge of one snge coeffcent c whch can be broadcasted by the recever In our case every reay has a separate power constrant Ths s a ore practca assupton n sensor networks snce every sensor or wreess devce has ts own battery power t The power contro soutons of the two cases are totay dfferent If reay seecton s used and ony one reay s aowed to cooperate t can be proved easy that we shoud choose the reay wth the hghest h hf g P P f g 1 f P 0 g P We ca h the reay seecton functon snce a reay wth a arger h resuts n a hgher receve SNR Whe a reays are aowed to cooperate the concepts of the best reay and reay seecton functon are not cear Snce the power contro probe s a couped one t s hard to easure how uch contrbuton a reay has As dscussed before n network beaforng a reay wth a arger φ does not necessary use a arger power or has ore contrbuton But we can concude that f φ k > φ the fracton of power used at Reay k α k s no ess than the fracton of power used at Reay α It s worth to enton that n network beaforng reays wth arge enough φ s use ther fu power no atter what ther fu power s Actuay t s not hard to see that f at one te channes of a reays are good every reay shoud use ts fu power V SIMULATION RESULTS In ths secton we show suated perforance of network beaforng and copare t wth perforance of other exstng schees Fgures and 3 show perforance of networks wth Rayegh fadng channes and the sae power constrant on the transtter and reays In other words f g are CN01 and P 0 P 1 P R P The horzonta axs of the fgures ndcates P In Fg suated bock error rates of network beaforng wth opta power contro are copared to those of best-reay seecton Larsson s schee [4] wth tota reay power P Aaout dstrbuted space-te codng [10] [40] and apfy-and-forward wthout power contro every reay uses ts axa power n a -reay network The nforaton sybo s s oduated as BPSK We can see that network beaforng outperfors a other schees It s about 05dB and db better than Larsson s Bock error rate Aaout DSTC Network beaforng Larsson s schee Best reay seecton AF wthout power contro P db Fg -reay network wth fadng channes and P P 0 Bock error rate AF wthout power contro Larsson s schee Network beaforng Best reay seecton P db Fg 3 3-reay network wth fadng channes and P P 0 schee and best-reay seecton respectvey Wth perfect channe knowedge at reays t s 7dB better than Aaout dstrbuted space-te codng whch needs no channe nforaton at reays Apfy-and-forward wth no power contro ony acheves dversty 1 dstrbuted space-te codng acheves a dversty sghty ess than two whe best-reay seecton network beaforng and Larsson s schee acheve dversty Fg 3 shows suated perforance of a 3-reay network under dfferent schees Sar dversty resuts are obtaned But for the 3-reay case network beaforng s about 15dB and 35dB better than Larsson s schee and bestreay seecton respectvey In Fg 4 we show perforance of a -reay network n whch P 0 P 1 P and P P/ That s the transtter and the frst reay have the sae power constrant whe the second reay has ony haf the power of the frst reay The channes are assued to be Rayegh fadng channes In Fg 5 we show perforance of a -reay network whose channes have both fadng and path-oss effects We assue that the dstance between the frst reay and the transtter/recever s 1 whe the dstance between the second reay and the transtter/recever s The path-oss exponent [39] s assued to be We aso assue that the transtter and

8 8 Bock error rate Fg 4 Bock error rate P db -reay network wth dfferent reay power AF wthout power contro Network beaforng Best reay seecton AF wthout power contro Network beaforng Larsson s schee n Best reay seecton anaytcay The souton can be obtaned wth a copexty that s near n the nuber of reays The power used at a reay depends on not ony ts own channes nonneary but aso a other channes n the network In genera t s not even a dfferentabe functon of channe coeffcents Suaton wth Rayegh fadng and path-oss channes show that network beaforng acheves fu dversty whe apfyand-forward wthout power contro acheves dversty 1 ony Network beaforng aso outperfors other cooperatve strateges We have ust scratched the surface of a brand-new area There are a ot of ways to extend and generaze ths work Frst t s assued n ths work that reays know ther channes perfecty whch s not practca n any networks Network beaforng wth ted and deayed feedback fro the recever s an portant ssue Second the reay network probed n ths paper has ony one par of transtter and recever When there are utpe transtter-and-recever pars an nterestng probe s how reays shoud aocate ther powers to ad dfferent councaton tasks Fnay the twohop protoco can be generazed as we For a gven network topoogy one reevant queston s how any hops shoud be taken to optze the crteron at consderaton for exape error rate or capacty VII ACKNOWLEDGMENT Ths work was supported n part by ARO under the Mut- Unversty Research Intatve MURI grant #W911NF Fg P db -reay network wth path-oss pus fadng channes reays have the sae power constrant e P 0 P 1 P P In both cases dstrbuted space-te codng does not appy and Larsson s schee appes for the second case ony So we copare network beaforng to best-reay seecton and apfy-and-forward wth no power contro ony Perforance of Larsson s schee s shown n Fg 5 as we Both fgures show the superorty of network beaforng to other schees VI CONCLUSIONS AND FUTURE WORK In ths paper we propose the nove dea of beaforng n wreess reay networks to acheve both dversty and array gan The schee s based on a two-step apfy-and-forward protoco We assue that each reay knows ts own channes perfecty Unke prevous works n network dversty the schee deveoped here uses not ony the channes phase nforaton but aso ther agntudes Match fters are apped at the transtter and reays durng the second step to cance the channe phase effect and thus for a coherent bea at the recever n the ean whe opta power contro s perfored based on the channe agntudes to decde the power used at the transtter and reays The power contro probe for networks wth any nubers of reays s soved REFERENCES [1] H Jafarkhan Space-Te Codng: Theory and Practce Cabrdge Acadec Press 005 [] A Hottnen O Trkkonen and R Wchan Mut-Antenna Transcever Technques for 3G and Beyond John Wey 003 [3] Y Chang and Y Hua Appcaton of space-te near bock codes to parae wreess reays n obe ad hoc networks n Prof of the 36th Asoar Conference on Sgnas Systes and Coputers Nov 003 [4] Y Hua Y Me and Y Chang Wreess antennas - akng wreess councatons perfor ke wrene councatons n Prof of IEEE AP-S Topca Conference on Wreess Councaton Technoogy Oct 003 [5] Y Tang and M C Vaent Coded transt acrodversty: bock space-te codes over dstrbuted antennas n Prof of IEEE Vehcuar Technoogy Conference 001-Sprng vo pp May 001 [6] A Sendonars E Erkp and B Aazhang User cooperaton dverstypart I: Syste descrpton IEEE Transactons on Councatons vo 51 pp Nov 003 [7] A Sendonars E Erkp and B Aazhang User cooperaton dverstypart II: Ipeentaton aspects and perforance anayss IEEE Transactons on Councatons vo 51 pp Nov 003 [8] R U Nabar H Böcske and F W Kneubuher Fadng reay channes: Perforance ts and space-te sgna desgn IEEE Journa on Seected Areas n Councatons pp Aug 004 [9] H Böcske R U Nabar O Oyan and A J Paura Capacty scang aws n MIMO reay networks IEEE Transactons on Wreess Councatons pp June 006 [10] Y Jng and B Hassb Dstrbuted space-te codng n wreess reay networks IEEE Trans on Wreess Co vo 5 pp Dec 006 [11] A F Dana and B Hassb On the power-effcency of sensory and ad-hoc wreess networks IEEE Transactons on Inforaton Theory pp Juy 006 [1] J N Lanean and G W Worne Dstrbuted space-te-coded protocos for expotng cooperatve dversty n wreess network IEEE Transactons on Inforaton Theory vo 49 pp Oct 003

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